Question

for each part below, use the figure to fill in the blank. if necessary, you may learn what the markings on a figure indicate. (a) find ( m angle v ). ( m angle v = square^circ ) (b) find ( m angle x ). ( m angle x = square^circ )

Explanation:

Part (a)

Step 1: Identify Triangle Type

Triangle \( \triangle VUT \) has all sides equal (\( VU = VT = UT = 12 \)), so it's equilateral.

Step 2: Equilateral Triangle Angles

In an equilateral triangle, all angles are \( 60^\circ \). Thus, \( m\angle V = 60^\circ \).

Part (b)

Step 1: Identify Triangle Type

Triangle \( \triangle XWY \) has two equal sides (marked with ticks), so it's isosceles with \( \angle X = \angle W \) or \( \angle X = \angle Y \)? Wait, the equal sides are \( XW \) and \( XY \)? Wait, no, the markings: \( XW \) and \( XY \)? Wait, the figure shows \( XW \) and \( XY \)? Wait, no, the triangle \( XWY \): sides \( XW \) and \( XY \)? Wait, no, the markings: \( XW \) has a tick, and \( XY \) has a tick? Wait, no, the problem says "the markings on a figure indicate". So in \( \triangle XWY \), sides \( XW \) and \( XY \) are equal? Wait, no, the angle at \( Y \) is \( 50^\circ \). Wait, no, in an isosceles triangle, the angles opposite equal sides are equal. Wait, the sides with ticks: \( XW \) and \( XY \)? Wait, no, the triangle is \( XWY \), with \( X \) at top, \( W \) at bottom left, \( Y \) at bottom right. The sides \( XW \) and \( XY \) have ticks? Wait, no, the side \( XW \) and \( XY \)? Wait, no, the side \( XW \) and \( WY \)? Wait, no, the problem's figure: \( XW \) has a tick, and \( XY \) has a tick? Wait, no, the user's figure: (b) shows triangle \( XWY \), with \( X \) connected to \( W \) (left side, tick), \( X \) connected to \( Y \) (right side, tick). Wait, no, the right side \( XY \) has a tick, and left side \( XW \) has a tick. So \( XW = XY \), so it's isosceles with \( \angle W = \angle Y \)? Wait, no, \( \angle Y \) is \( 50^\circ \). Wait, no, in triangle \( XWY \), sides \( XW \) and \( XY \) are equal, so angles opposite: \( \angle W \) and \( \angle Y \). Wait, \( \angle Y = 50^\circ \), so \( \angle W = 50^\circ \). Then \( \angle X = 180 - 50 - 50 = 80^\circ \). Wait, no, wait: the sum of angles in a triangle is \( 180^\circ \). So if \( \triangle XWY \) is isosceles with \( XW = XY \), then angles at \( W \) and \( Y \) are equal? Wait, no, \( XW = XY \), so the angles opposite: \( \angle Y \) and \( \angle W \). So \( \angle Y = \angle W = 50^\circ \), then \( \angle X = 180 - 50 - 50 = 80^\circ \). Wait, but let's confirm:

Step 1: Identify Isosceles Triangle

In \( \triangle XWY \), sides \( XW \) and \( XY \) are equal (marked with ticks), so it's isosceles with \( \angle W = \angle Y \).

Step 2: Sum of Angles

Sum of angles in a triangle: \( 180^\circ \). So \( m\angle X + m\angle W + m\angle Y = 180^\circ \). Given \( m\angle Y = 50^\circ \), so \( m\angle W = 50^\circ \) (since \( \angle W = \angle Y \)). Then \( m\angle X = 180 - 50 - 50 = 80^\circ \).

Answer:

(a) \( m\angle V = \boxed{60}^\circ \)
(b) \( m\angle X = \boxed{80}^\circ \)